All Questions
15,509 questions
2
votes
1
answer
177
views
Optimization over Poisson-binomial distributions
I am studying the problem of how an expected utility maximizer should optimally form a portfolio of uncorrelated Bernoullis.
Fix an increasing sequence of $n$ numbers in $(0,1)$, $0<p_1<\dots<...
2
votes
0
answers
119
views
gcrd and associates of an element of the quaternion algebra over a totally real number field $K$
Let $K$ be a totally real number field of class number 1, and $Q$ the quaternion algebra over the ring of integers of $K$ with basis
$\{1,i,j,k\}$ such that $i^2 = j^2 = k^2 = -1$ and $ij = -ji, ik = ...
- 133
2
votes
1
answer
170
views
Law of large numbers for a continuum of Bernoullis
Suppose I have a family of $n$ independent Bernoulli random variables described by a vector of parameters $(p_i)_{i=1}^n$. As it is well known, the number of successes within this family is a random ...
2
votes
1
answer
197
views
Seeking articles on closed-form formulas for specific partial fraction expansions
I'm currently researching a general closed-form formula, in terms of elementary functions, for functions that have the following type of partial fraction expansion:
$$\frac{1}{x^{p}}+\sum_{n=1}^{+\...
- 463
8
votes
0
answers
156
views
Square root of an Anosov diffeomorphism
Let $T\colon \mathbb T^d\to\mathbb T^d$ be an Anosov diffeomorphism (that is, the tangent bundle splits into invariant stable and unstable bundles; the restriction of $DT$ to the unstable bundle is ...
- 23.2k
1
vote
0
answers
29
views
Most general filtered algebras with Hilbert polynomials and multiplicities
Let $k$ be any base field and $A$ an affine infinite dimensional $k$-algebra.
Let $\mathcal{F}= \{ A_i \}_{i \geq 0}$ be a finite dimensional filtration for $A$: that is, $k \subset A_0$ and each $A_i$...
- 3,328
9
votes
2
answers
499
views
On the $\phi^4$-model on infinite lattice
It is mentioned in this answer Is there a program to solve The Yang–Mills Existence and Mass Gap problem similar to the Hamilton's program to solve Poincaré Conjecture?
that it is an open ...
- 577
3
votes
0
answers
152
views
My category is rigid: what this implies for representation theory?
I am studying a subcategory $\mathcal{C}$ of modules for an associative noncommutative algebra $A$ (which is in fact also a Hopf algebra).
It is clear from our definition of $\mathcal{C}$ that it is ...
- 3,328
4
votes
0
answers
335
views
Book recommendation in functional analysis and probability
I am interested by functional analysis and probability. I would like to know if you have any books that deal with these two subjects (at a graduate level) to recommend?
I'm looking for a book that has ...
1
vote
0
answers
49
views
Boundary-condition-changing Operators for Free Boson BCFT with Dirichlet Boundary Conditions (or more general BCFTs)?
(NOTE: This is a crosspost from this Physics.SE post)
Is there any literature about boundary-condition-changing (b.c.c.) operators for the Free Boson with Dirichlet Boundary Conditions? The b.c.c. ...
- 545
2
votes
0
answers
681
views
Roadmap for p-adic geometry
I think some questions asked in similar fashion with this one. I am a master student in mathematics. I have knowledge in algebraic geometry(both in Shafarevich's and Vakil's books), algebraic topology ...
- 21
2
votes
1
answer
90
views
Heat kernel convergence when expanding domains
Let $\Omega$ be an arbitrary domain in $\mathbb{R}^n$. There exists a positive $C^{\infty}$ function $G_{\Omega} : \Omega \times \Omega \times (0, \infty) \rightarrow \mathbb{R}$ (Dirichlet heat ...
- 677
12
votes
2
answers
837
views
Restriction of $\mathrm{GL}(n)$ representation to $S_n$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\O{O}$I'm looking for a reference to cite for the following observation. Given an irreducible representation of $\GL(n)$ labelled by the Young diagram ...
- 21.6k
3
votes
1
answer
209
views
Pathwise Hölder continuity of Ito diffusions - is this result written anywhere?
Let $X$ be the solution to the multidimensional SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $W$ a Brownian motion, and $\mu, \sigma$ Lipschitz continuous with $\sigma$ nowhere zero. I'm ...
- 6,323
4
votes
1
answer
175
views
Viscosity solutions of eikonal equation on Riemannian manifolds
It is well known that given a bounded open region $\Omega \subset \mathbb{R}^n$, the Dirichlet problem $$\lVert \nabla u \rVert = 1, \quad u|_{\partial \Omega} = 0$$
admits the unique viscosity ...
- 235
16
votes
1
answer
358
views
Galois cohomology for non-Galois extensions
If $L/K$ is a Galois extension with group $G$ then we can consider $H^*(G;L^\times)$. This is useful in algebraic number theory, and there are many results about it.
Now let $L/K$ be a finite ...
- 56.9k
3
votes
1
answer
212
views
A few points of clarification on the Martin boundary
Let $\Gamma$ be a finitely generated group, and let $M$ be the Martin boundary of $\Gamma$. I was reading the article on Martin boundary on Encyclopedia of Math, and I have a few questions about what ...
- 1,407
3
votes
0
answers
171
views
Nice blowups are universal algebraic fiber spaces?
We say that a proper (maybe projective) morphism $f:X \to Y$ is a universal algebraic fiber space if $f_* O_X = O_Y$ holds universally. (This means: for any morphism $Y' \to Y$, if $X' = Y' \times_Y X$...
- 39
3
votes
0
answers
219
views
Schwartz's theorem without English language reference
I'm reading the paper "Spectral Synthesis And The Pompeiu Problem" by Leon Brown, Bertram M. Schreiber and B. Alan Taylor,
Annales de l’Institut Fourier 23, No. 3, 125-154 (1973), MR352492, ...
- 421
8
votes
0
answers
341
views
Has the notion of a unipotent group scheme been studied?
The concept of a unipotent algebraic group over a field has been extensively studied and is fundamental in algebraic geometry. However, has the notion of a unipotent group scheme over a general base ...
- 773
2
votes
1
answer
148
views
Is projection of a closed subspace Borel?
Specifically, letting $E$ be a separable infinite-dimensional real Banach space, and $D_2$ in $E\times E$ a closed linear subspace, is then $\{\,x:\exists\,y\,;(x,y)\in D_2\}$ a Borel set in $E\,$? ...
- 3,584
7
votes
1
answer
238
views
What is a Whitney Jet?
I'm currently reading Michor, Manifold of Mappings for Continuum Mechanics. In this paper he makes use of 'Whitney Jets' but takes it to be an already understood concept. I'm familiar with jets but ...
- 2,369
4
votes
1
answer
686
views
Who and when proved Artin's Theorem on alternative rings?
I am interested in the history of the proof of Artin's Theorem (on the diassociativity of alternative rings).
Question. When has Artin proved this theorem and where was it published for the first ...
- 42k
2
votes
1
answer
81
views
Rate of convergence of random samples wrt Hausdorff distance
Let $X$ be a compact metric space with a probability measure $\mu$. We can draw random samples $X_n = \{x_1,\cdots, x_n\}$ from $X$ using $\mu$, and I am interested in the rate of convergence of $X_n$ ...
- 305
16
votes
3
answers
4k
views
Is it known that the Collatz-like sequence with 7n+1 diverges to infinity starting with 7?
In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...
- 541
1
vote
0
answers
92
views
The existence of such homomorphism [closed]
Are there any papers or books that investigate/discuss the relationship between conjugacy classes and normality for the existence of non-trivial homomrphism f:G->H were H is some nontrivial ...
- 61
2
votes
0
answers
119
views
What are the finite-dimensional irreducible unitary representations of $E(3)$?
Let $E(3)$ be the Euclidean group of $\mathbb{R}^3$ defined, e.g., by
$$E(3)=SO(3)\ltimes T(3)$$
where $T(3)$ is the translation group.
I am looking for a reference classifying all the finite-...
- 31
4
votes
0
answers
116
views
Do any Legendrian knots in standard contact 3-space have big tubular neighborhoods?
Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$.
According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular ...
- 501
7
votes
1
answer
432
views
Strict toposes as a finite limit theory
For some motivation I have been wondering about generalizing the topos of coalgebras theorem to relative monads in my previous question. This brought me to wonder about topos objects.
NLab on ...
- 1,347
3
votes
1
answer
162
views
Counting equal covering sets
Definition. We call a set $C$ of sets to be an equal covering set of $S$ if the elements of $C$ are all the same size and each element of $S$ is contained an equal number of times throughout the sets ...
- 33
4
votes
0
answers
177
views
Recording of 2009 lecture on Harvey Friedman's work
On December 13--20 2009 at Bristol, there was a meeting devoted to thorough dissection of Harvey Friedman's work on the foundations of mathematics and his statements claimed to be equivalent to ...
- 2,031
2
votes
0
answers
177
views
Eigenspaces of complex conjugation on étale cohomology of a smooth projective curve
Let $X$ denote a smooth projective curve defined over $\mathbb{Z}[1/N]$, and its base change $ \overline{X} $ to $ \overline{\mathbb{Q}} $. Let $ V $ be a $ p $-adic local system on $X$ ($p\mid N$), ...
- 2,907
6
votes
1
answer
275
views
Reference: the category of derived affine schemes is extensive
The category (that is, $(\infty, 1)$-category) of derived affine schemes is the opposite category of the localization of simplicial commutative rings in weak equivalences.
See extensive category. Does ...
- 6,962
11
votes
3
answers
728
views
Application of Lie group analysis of PDE (beyond calculation of exact solutions)
I am learning the Lie symmetry group method for PDEs. In my reading, all of the applications of this method are to calculate the exact solutions of PDEs. Are there any good references which provide ...
- 111
2
votes
0
answers
49
views
Deformed preprojective algebras of generalized Dynkin type
Question 1:Is it true that the selfinjective (finite dimensional over an algebraically closed field K) algebras $A$ such that the stable module category of $A$ is 2-Calabi-Yau are exactly the deformed ...
- 26.5k
2
votes
1
answer
182
views
Reference request: uniformization theorem proof by Borel
This answer refers to a proof of the uniformization theorem via the PDE describing metrics of constant curvature (Liouville?) by Borel. I haven’t been able to find this reference, is anyone aware ...
- 661
34
votes
3
answers
5k
views
A trigonometric equation: how hard could it be?
The following problem started out with a formulation in terms of complex numbers: let $\epsilon=e^{\frac{\pi i}3}$ and $z=e^{\frac{2\pi i}{3(2n-1)}}$. It's rather amusing that the following appears to ...
- 43.2k
2
votes
1
answer
98
views
Locally compact groupoid with a Haar system such that the range map restricted to isotropy groupoid is open
Can somebody provide an example of a locally compact groupoid $G$ with a Haar system such that the range map restricted to isotropy groupoid of $G$ is open?
I could not find any specific example for ...
19
votes
6
answers
2k
views
Book recommendation introduction to model theory
Next semester I will be teaching model theory to master students. The course is designed to be "soft", with no ambition of getting to the very hardcore stuff. Currently, this is the syllabus....
- 9,111
1
vote
1
answer
232
views
Looking for q-analog of derangement anagrams for a word
I have already known QPermutationDerangement:
It describes the distribution
$$
d_n(q)=\sum_{\sigma \in D_n} q^{\operatorname{maj}(\sigma)}
$$
Where we sum over all derangements of an $n$ element set.
...
- 175
2
votes
0
answers
101
views
Majorization theory on $\sigma$-finite measure spaces
I want to learn about majorization and submajorization theory on $\sigma$-finite measure spaces. I know things get a bit more complicated compared with the case of a finite measure spaces but I'm ...
- 769
2
votes
1
answer
111
views
Structure of elements of a finite group not contained in any conjugate of a proper subgroup
Let $G$ be a finite group and $H$ be a proper subgroup of $G$. It is elementary to prove that the union of all conjugates of $H$ under $G$,
$$U:=\bigcup_{\sigma\in G}\sigma^{-1}H\sigma,$$
is properly ...
10
votes
1
answer
516
views
Earliest proof of Solovay's theorem for successor cardinals
Solovay's partition theorem states that a stationary set over a regular cardinal $\kappa$ can be partitioned into $\kappa$-many disjoint stationary sets. The full theorem was proven by Solovay in 1971 ...
- 576
1
vote
1
answer
114
views
Identifying player strategies in repeated games, based on payoffs
Background
In evolutionary game theory, one can what kinds of different strategies yield the most payoff to players that play the same game repeatedly. Consider, for instance, the iterated Prisoner's ...
- 4,829
9
votes
1
answer
247
views
Is the standard model structure on reduced simplicial sets cofibrantly generated?
Let $\mathrm{sSet}_0$ be the category of simplicial sets with a single zero cell, also known as reduced simplicial sets. It is a well known fact (due to Quillen) that $\mathrm{sSet}_0$ supports a ...
- 10.9k
6
votes
0
answers
145
views
Alternative proofs of the countable chain condition in forcing
Advance warning: This question is more about history and pedagogy than "hard" mathematics.
I am studying Cohen forcing with the forcing poset $(\operatorname{Fin}(E,2),\supseteq,0)$, and I ...
6
votes
1
answer
274
views
Subrepresentations of the $\text{SL}_n(k)$-representation $\mathfrak{gl}_n(k)$
$\DeclareMathOperator\SL{SL} \newcommand{\gl}{\mathfrak{gl}} \newcommand{\sl}{\mathfrak{sl}}$One of my graduate students asked me for a reference for the following fact. Let $k$ be a general field (...
- 44.9k
4
votes
0
answers
184
views
Étale- or fppf-crystalline sites
I have a straightforward question. Let (say) $X/\mathbb{F}_p$ be a smooth proper scheme. On the big crystalline category over $\mathbb{Z}/p^n$ one can take the Zariski or étale topology, and one can ...
- 371
5
votes
2
answers
376
views
Non-orientable real algebraic three-dimensional manifolds
Smooth real algebraic hypersurfaces of even degree in $\mathbb{RP}^4$ that are maximal (i.e. that are homologically as rich as possible in the sense of the Smith-Thom inequality) are all non-...
- 183
1
vote
0
answers
143
views
Explicit computation of Čech-cohomology of coherent sheaves on $\mathbb{P}^n_A$
$\newcommand{\proj}[1]{\operatorname{proj}(#1)}
\newcommand{\PSP}{\mathbb{P}}$These days I noticed the following result of (constructive) commutative algebra, which I think is probably well known ...
- 708